
\magnification = 2200          % scale all non "true" dimensions by a factor of 2.2

\hsize 6.52 true in                       % width of page  (not scaled because of the true
\vsize 9.2 true in
\hoffset = -0.1 true in
\voffset -0.25 true in
\parskip= 1 true pt


\def\lf{\ \hfil\break}                  % This defines \lf to mean a newline
\def\cl{\centerline}                  % used to create a centered line
\def\LF{\medskip\noindent}    % makes \LF mean a newline plus some vertical space


\nopagenumbers                  % Just what it says  :-)


\cl{\bf              About the Lopez-Ros No-Go Theorem}

\vskip 10pt

\cl{                               H. Karcher                         }
\vskip 10pt

\lf
    The  Lopez-Ros Theorem [LR] says that a  complete,
minimal embedding of a punctured sphere is either a Catenoid
or a plane. Our example is parametrized by a 3-punctured sphere.
Its Gauss map is $\hbox{Gauss}(z) := cc (z-1)(z+1).$
The differential $dh = (z^2-1)/(z^2 - ee)^2$  puts the punctures at
$ +ee$, $-ee$ and $\infty$. Parameter lines on the sphere extend
polar coordinates around the punctures at $z=+ee$, $z=-ee$,
$z= \infty$ in order to make the ends look nice.

\lf
   A necessary condition for embeddedness  is that the normals
of all ends are parallel, i.e.,   $ee=1$.  In this case a residue
computation shows that  the period cannot be closed, in agreement
with the  theorem of Lopez-Ros.
If $ee >1$, then $cc$ can be chosen to close the period, but then
the catenoid  ends are tilted so that they intersect the third
(planar) end.
The default morph in 3DXM shows what happens when $ee$
approaches $1$ while the periods are always closed (with a closing
value of $cc$ that grows to infinity). In a properly scaled
picture the surface looks more and more like two catenoids at
larger and larger distance.
\lf
Since we also want to show morphs were the period opens up, the program
uses $cc$ as follows: If the user sets $cc = 0$ then the program
recomputes $cc$ to the period-closing value $cc_{close}$. Otherwise we
only restrict $cc$ by resetting it as \lf\phantom{.}
$\hskip 3mm cc = \max(0.5*cc_{close}, \min(cc, 2*cc_{close}))$.
\lf  To choose one's own morph, 
note that the surface has a gap if $cc$ is larger than $cc_{close}$, 
and that  it intersects itself if $cc$ is smaller than  $cc_{close}$.
As an example, compute first with $ee=1.01, cc=0$; then, in the
Set Morphing Dialog, click the button  {\it Initialize to current parameters}, and finally
morph $ee$ from the current value $ee0=1.01$ to some larger value, e.g.
$ee1=1.08$.

\bigskip
\cl{\bf References}
\lf
[LR]  F.J. Lopez and A. Ros,  On embedded complete minimal
        surfaces of genus zero, Journal of Differential Geometry
        33 (1), 1991, pp 293--300.

\bigskip
\goodbreak
\lf
   For a discussion of techniques for creating minimal surfaces with
various qualitative features by appropriate choices of Weierstrass
data, see either [KWH], or pages 192--217 of [DHKW].

\lf
[KWH]  H. Karcher, F. Wei, and D. Hoffman, The genus one helicoid, and
          the minimal surfaces that led to its discovery, in ``Global Analysis
          in Modern Mathematics, A Symposium in Honor of Richard Palais'
          Sixtieth Birthday'', K. Uhlenbeck Editor, Publish or Perish 
Press, 1993

\lf
[DHKW] U. Dierkes, S. Hildebrand, A. Kuster, and O. Wohlrab,
            Minimal Surfaces I, Grundlehren der math. Wiss. v. 295
            Springer-Verlag, 1991


 \end{document}